{"id":792,"date":"2025-10-09T15:39:33","date_gmt":"2025-10-09T15:39:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=792"},"modified":"2025-12-15T13:59:31","modified_gmt":"2025-12-15T13:59:31","slug":"stokes-theorem-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/stokes-theorem-apply-it\/","title":{"raw":"Stokes\u2019 Theorem: Apply It","rendered":"Stokes\u2019 Theorem: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use Stokes' theorem to relate line integrals to surface integrals<\/li>\r\n \t<li>Use Stokes\u2019 theorem to calculate a surface integral<\/li>\r\n \t<li>Use Stokes\u2019 theorem to calculate a curl<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Analyzing Electromagnetic Induction in a Wire Loop<\/h2>\r\nYou are learning electronics repair at a local makerspace and currently studying electromagnetic induction\u2014the principle that makes transformers, generators, and wireless chargers work. Your mentor has set up a demonstration with a wire loop placed in a changing magnetic field, and you need to analyze the relationship between the electric field induced in the wire and the magnetic flux through the loop. Understanding how Stokes' theorem connects the circulation of the electric field around the wire to the curl of the field over the surface helps you predict voltage outputs and troubleshoot faulty coils. Today, you will work through several calculations involving vector fields, verifying relationships between line integrals and surface integrals, and applying Faraday's law to determine how electric fields respond to changing magnetic fields.\r\n\r\nConsider a vector field representing a simplified electric field model: [latex]{\\bf{F}}(x,y,z)=\\langle 2z, x, y \\rangle[\/latex]. You need to verify Stokes' theorem by computing both the line integral around a boundary curve and the surface integral of the curl over a surface.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1] 317061[\/ohm_question]<\/section>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Your mentor presents a more complex surface: a wire loop forms the boundary of a paraboloid cap. Consider vector field [latex]{\\bf{F}}(x,y,z) = \\langle y, -x, z^2 \\rangle[\/latex] and surface [latex]S[\/latex], which is the part of the paraboloid [latex]z = 9 - x^2 - y^2[\/latex] that lies above the [latex]xy[\/latex]-plane, oriented upward. The boundary curve [latex]C[\/latex] is the circle [latex]x^2 + y^2 = 9[\/latex] in the [latex]xy[\/latex]-plane.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rather than computing a complicated surface integral, use Stokes' theorem to evaluate [latex]\\displaystyle\\iint_S \\text{curl }{\\bf{F}} \\cdot d{\\bf{S}}[\/latex] by instead computing the line integral [latex]\\displaystyle\\int_C {\\bf{F}} \\cdot d{\\bf{r}}[\/latex].<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317062 [\/ohm_question]<\/section>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An important consequence of Stokes' theorem is surface independence: if two surfaces share the same boundary curve and orientation, then the flux integral of the curl is the same over both surfaces.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Consider vector field [latex]{\\bf{G}}(x,y,z) = \\langle z, x, y \\rangle[\/latex]. Surface [latex]S_1[\/latex] is the hemisphere [latex]z = \\sqrt{1 - x^2 - y^2}[\/latex], and surface [latex]S_2[\/latex] is the flat disk [latex]x^2 + y^2 \\leq 1[\/latex] in the plane [latex]z = 0[\/latex]. Both surfaces are oriented upward and share the same boundary: the unit circle in the [latex]xy[\/latex]-plane.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317064 [\/ohm_question]<\/section>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Now your mentor explains Faraday's law of electromagnetic induction. In the makerspace, you're examining a coil where the magnetic field is changing over time.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The magnetic field is given by [latex]{\\bf{B}}(t) = \\langle 0, 0, 5t^2 \\rangle[\/latex], where [latex]t[\/latex] represents time in seconds. According to Faraday's law in differential form, [latex]\\text{curl }{\\bf{E}} = -\\frac{\\partial {\\bf{B}}}{\\partial t}[\/latex].<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1] 317065[\/ohm_question]<\/section>Your mentor shows you a scenario where the magnetic field is constant: [latex]{\\bf{B}} = \\langle 2, -3, 7 \\rangle[\/latex]. This represents a static magnetic field that does not change over time.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317066 [\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use Stokes&#8217; theorem to relate line integrals to surface integrals<\/li>\n<li>Use Stokes\u2019 theorem to calculate a surface integral<\/li>\n<li>Use Stokes\u2019 theorem to calculate a curl<\/li>\n<\/ul>\n<\/section>\n<h2>Analyzing Electromagnetic Induction in a Wire Loop<\/h2>\n<p>You are learning electronics repair at a local makerspace and currently studying electromagnetic induction\u2014the principle that makes transformers, generators, and wireless chargers work. Your mentor has set up a demonstration with a wire loop placed in a changing magnetic field, and you need to analyze the relationship between the electric field induced in the wire and the magnetic flux through the loop. Understanding how Stokes&#8217; theorem connects the circulation of the electric field around the wire to the curl of the field over the surface helps you predict voltage outputs and troubleshoot faulty coils. Today, you will work through several calculations involving vector fields, verifying relationships between line integrals and surface integrals, and applying Faraday&#8217;s law to determine how electric fields respond to changing magnetic fields.<\/p>\n<p>Consider a vector field representing a simplified electric field model: [latex]{\\bf{F}}(x,y,z)=\\langle 2z, x, y \\rangle[\/latex]. You need to verify Stokes&#8217; theorem by computing both the line integral around a boundary curve and the surface integral of the curl over a surface.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317061\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317061&theme=lumen&iframe_resize_id=ohm317061&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Your mentor presents a more complex surface: a wire loop forms the boundary of a paraboloid cap. Consider vector field [latex]{\\bf{F}}(x,y,z) = \\langle y, -x, z^2 \\rangle[\/latex] and surface [latex]S[\/latex], which is the part of the paraboloid [latex]z = 9 - x^2 - y^2[\/latex] that lies above the [latex]xy[\/latex]-plane, oriented upward. The boundary curve [latex]C[\/latex] is the circle [latex]x^2 + y^2 = 9[\/latex] in the [latex]xy[\/latex]-plane.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rather than computing a complicated surface integral, use Stokes&#8217; theorem to evaluate [latex]\\displaystyle\\iint_S \\text{curl }{\\bf{F}} \\cdot d{\\bf{S}}[\/latex] by instead computing the line integral [latex]\\displaystyle\\int_C {\\bf{F}} \\cdot d{\\bf{r}}[\/latex].<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317062\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317062&theme=lumen&iframe_resize_id=ohm317062&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An important consequence of Stokes&#8217; theorem is surface independence: if two surfaces share the same boundary curve and orientation, then the flux integral of the curl is the same over both surfaces.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Consider vector field [latex]{\\bf{G}}(x,y,z) = \\langle z, x, y \\rangle[\/latex]. Surface [latex]S_1[\/latex] is the hemisphere [latex]z = \\sqrt{1 - x^2 - y^2}[\/latex], and surface [latex]S_2[\/latex] is the flat disk [latex]x^2 + y^2 \\leq 1[\/latex] in the plane [latex]z = 0[\/latex]. Both surfaces are oriented upward and share the same boundary: the unit circle in the [latex]xy[\/latex]-plane.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317064\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317064&theme=lumen&iframe_resize_id=ohm317064&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Now your mentor explains Faraday&#8217;s law of electromagnetic induction. In the makerspace, you&#8217;re examining a coil where the magnetic field is changing over time.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The magnetic field is given by [latex]{\\bf{B}}(t) = \\langle 0, 0, 5t^2 \\rangle[\/latex], where [latex]t[\/latex] represents time in seconds. According to Faraday&#8217;s law in differential form, [latex]\\text{curl }{\\bf{E}} = -\\frac{\\partial {\\bf{B}}}{\\partial t}[\/latex].<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317065\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317065&theme=lumen&iframe_resize_id=ohm317065&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Your mentor shows you a scenario where the magnetic field is constant: [latex]{\\bf{B}} = \\langle 2, -3, 7 \\rangle[\/latex]. This represents a static magnetic field that does not change over time.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317066\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317066&theme=lumen&iframe_resize_id=ohm317066&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-792","chapter","type-chapter","status-publish","hentry"],"part":438,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/792","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/792\/revisions"}],"predecessor-version":[{"id":2806,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/792\/revisions\/2806"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/438"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/792\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=792"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=792"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=792"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=792"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}